Subgroups — C₂×#22562118 #31009319

Order 17152.

542 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562118	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863172	C₂	1	2	yes	maximal
5	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	8	yes	—
11	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
12	#16863172	C₂²	1	8	yes	—
13	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
14	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₄	1	4	yes	—
16	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	4	no	—
23	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
24	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
26	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
27	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₆₇	C₈	1	4	yes	—
32	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₈	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
38	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₆₇	C₂³	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	4	yes	—
43	C₆₇⋊C₁₆	—	2	2	no	—
44	C₆₇⋊C₁₆	—	2	4	no	—
45	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
46	C₂×Q₈×C₆₇	—	2	4	no	—
47	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	4	no	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₆₇	#41	1	2	yes	—
60	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
62	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
63	C₄²×C₆₇	C₂×C₈	1	4	yes	—
64	C₄²×C₆₇	C₂×C₈	1	2	yes	—
65	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
66	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
68	Q₈×C₆₇	—	2	8	no	—
69	C₂×C₄×C₆₇	—	2	4	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
73	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
75	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₆₇	#131	1	2	yes	—
80	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₈×C₆₇	—	2	2	no	—
82	C₈×C₆₇	—	2	4	no	—
83	C₈×C₆₇	C₄×Q₈	1	2	yes	—
84	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
86	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
87	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
88	C₂²×C₆₇	#504	1	2	yes	—
89	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₆₇	—	2	4	no	—
91	C₄×C₆₇	—	2	4	no	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	C₈×Q₈	1	2	yes	—
94	C₄×C₆₇	C₂×M₃₂	1	4	yes	—
95	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
96	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
97	C₂×#2183	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	#2183	1	4	yes	complemented
99	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
100	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
101	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
102	C₂×C₈×Q₈	D₆₇	1	1	yes	characteristic
103	#2183	—	67	8	no	—
104	C₂×C₄⋊C₁₆	—	67	1	no	—
105	C₂×C₄⋊C₁₆	—	67	2	no	—
106	C₂×#346	—	67	2	no	—
107	C₂×C₄×C₁₆	—	67	1	no	—
108	C₆₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
110	C₂²×C₁₆	—	67	1	no	—
111	C₂²×C₁₆	—	67	1	no	—
112	C₂²×C₁₆	—	67	2	no	—
113	C₈×Q₈	Dic₆₇	1	4	yes	—
114	C₈×Q₈	C₂×D₆₇	1	4	yes	—
115	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
116	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
119	C₄⋊C₁₆	—	67	4	no	—
120	C₄⋊C₁₆	—	67	8	no	—
121	#346	—	67	8	no	—
122	C₄×C₁₆	—	67	4	no	—
123	C₂²×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
126	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	C₄×Q₈	C₆₇⋊C₈	1	4	yes	—
128	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
129	C₂×C₄⋊C₄	C₆₇⋊C₈	1	2	yes	—
130	C₂×C₄⋊C₄	C₆₇⋊C₈	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
132	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
133	C₂×C₁₆	—	67	4	no	—
134	C₂×C₁₆	—	67	2	no	—
135	C₂×C₁₆	—	67	2	no	—
136	C₂×C₁₆	—	67	4	no	—
137	C₂×C₁₆	—	134	2	no	—
138	C₂×C₁₆	—	134	4	no	—
139	C₄⋊C₈	C₂×Dic₆₇	1	8	yes	—
140	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
141	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
142	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
143	C₄×C₈	C₂²×D₆₇	1	2	yes	—
144	C₄×C₈	C₂²×D₆₇	1	4	yes	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
147	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
148	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
150	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
155	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
156	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
157	C₂×C₈	Q₈×D₆₇	1	2	yes	—
158	C₂×C₈	#11761629	1	2	yes	—
159	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
161	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
162	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
163	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
164	C₄²	C₂²×Dic₆₇	1	4	yes	—
165	C₄²	C₂²×Dic₆₇	1	2	yes	—
166	C₁₆	—	67	4	no	—
167	C₁₆	—	134	2	no	—
168	C₁₆	—	134	4	no	—
169	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
170	Q₈	—	2	8	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	#14523983	1	4	yes	—
175	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
176	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
177	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
178	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
179	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
181	C₂×C₄	#14524050	1	2	yes	—
182	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	4	no	—
185	C₈	Q₈×Dic₆₇	1	2	yes	—
186	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
187	C₈	C₂×#11761629	1	1	yes	characteristic
188	C₂²	Q₈×C₆₇⋊C₈	1	2	yes	—
189	C₂²	C₂×#14523983	1	2	yes	—
190	C₂²	#16863813	1	2	yes	—
191	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	Q₈×C₆₇⋊C₈	1	2	yes	—
196	C₄	C₂×#14523983	1	4	yes	—
197	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
198	C₄	C₂×#14524050	1	1	yes	characteristic
199	C₂	#22562118	1	4	yes	direct product factor
200	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
201	C₂	C₂²×#14523983	1	1	yes	characteristic
202	C₂	C₂×#16863813	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₈×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₆₇
hypocenter	C₆₇	C₂×#2183
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22562118	C₂	direct (N × H)
23. C₂×C₆₇⋊C₁₆	Q₈	semidirect
45. C₆₇⋊C₁₆	C₂×Q₈	semidirect
98. C₂×C₆₇	#2183	semidirect
108. C₆₇	C₂×#2183	semidirect
199. C₂	#22562118	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22562118	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×#16863172	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×C₈×Q₈×C₆₇	2
C₂×#2183	67

Subgroups — C2×#22562118 #31009319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562118 #31009319